1. **State the problem:** Calculate the value of the expression $$\frac{(0.0050)^2 \cdot (0.8)^3}{(4 \times 25.783 \times 10^{-3})(5.013 \times 10^{-3})^3} \times (7.715 \times 10^{-6})^2$$. 2. **Rewrite the expression clearly:** $$\frac{(0.0050)^2 \cdot (0.8)^3}{(4 \times 25.783 \times 10^{-3}) \cdot (5.013 \times 10^{-3})^3} \times (7.715 \times 10^{-6})^2$$ 3. **Calculate each part step-by-step:** - Calculate $(0.0050)^2$: $$ (0.0050)^2 = 0.0050 \times 0.0050 = 0.000025 $$ - Calculate $(0.8)^3$: $$ (0.8)^3 = 0.8 \times 0.8 \times 0.8 = 0.512 $$ - Calculate the numerator: $$ 0.000025 \times 0.512 = 0.0000128 $$ - Calculate the denominator part 1: $$ 4 \times 25.783 \times 10^{-3} = 4 \times 0.025783 = 0.103132 $$ - Calculate $(5.013 \times 10^{-3})^3$: $$ (5.013 \times 10^{-3})^3 = 5.013^3 \times (10^{-3})^3 = 125.754 \times 10^{-9} = 1.25754 \times 10^{-7} $$ - Calculate the denominator: $$ 0.103132 \times 1.25754 \times 10^{-7} = 1.296 \times 10^{-8} $$ - Calculate the fraction: $$ \frac{0.0000128}{1.296 \times 10^{-8}} = \frac{1.28 \times 10^{-5}}{1.296 \times 10^{-8}} = \frac{1.28}{1.296} \times 10^{3} $$ - Simplify the fraction: $$ \frac{1.28}{1.296} = \cancel{\frac{1.28}{1.296}} \approx 0.9877 $$ - So the fraction is approximately: $$ 0.9877 \times 10^{3} = 987.7 $$ - Calculate $(7.715 \times 10^{-6})^2$: $$ 7.715^2 \times (10^{-6})^2 = 59.54 \times 10^{-12} = 5.954 \times 10^{-11} $$ 4. **Multiply the fraction by the last term:** $$ 987.7 \times 5.954 \times 10^{-11} = 5.877 \times 10^{-8} $$ 5. **Final answer:** $$ \boxed{5.88 \times 10^{-8}} $$ This is the value of the given expression calculated step-by-step.